Question: Can an argument be fallacious for liberals yet sound for conservatives, and vice versa?

Just a quick correction, I think you meant to switch "valid" and "sound". An argument is only sound if the premises are true, so one's proclivity to accept the truth of premises would affect the argument's "soundness" (in their minds). All sound arguments must be valid, but all valid argument don't need to be sound.

Let's clean this up just to ensure that it is a deductive argument rather than an informal one:

P1. Immigrants pose a financial drain on the US economy.
P2. Our economy suffers with from a financial drain of this magnitude.
P3. Therefore, we cannot if we allow immigrants to enter the country our economy will suffer.

A few things to note:

1) Removed language that introduces ambiguity (e.g., "of this magnitude') - it isn't necessary
2) The conclusion, as it was originally written, came to a broad conclusion not supported by the premises. What if the good immigrants do outweigh the financial drain on the economy?
3) Regardless of one's political, religious, or other ideological position, the truth isn't affected. Like the saying goes, facts don't care if you believe in them or not; they're still facts. This doesn't mean that one can't believe an argument to be sound when it really isn't. Then the argument shifts from the validation of the argument to the validation of the (alleged) facts in the premises.

Hope that answers your question!